Why can't I loop cut all the way around the object?
If the mesh already is triangulated you could use the Knife tool (K) in Cut Through (Z) and Angle Constraint (C) Mode when you are in a side orthographical view. This will let you add edges regardless of the topology.

But most of the time it's better to just un-triangulate it manually by Dissolve edge or with Tris to Quads (Alt-J) in the face menu (CTRL-F). 

If you have ngons you can find them with the Select All by Trait/Faces by Sides and set the number of vertices to Grater than 4. 

You will find it much easier to work with quads.